Question

A helium-neon laser beam has a wavelength in air of 633 nm. It takes 1.38 ns for the light to travel through 30 cm of an unknown liquid. What is the wavelength of the laser beam in the liquid?

Short answer

The wavelength of the laser beam through the unknown liquid is,$458.7nm.$

Step-by-step solution

Given data

Helium-neon laser beam has a wavelength in air of ${\lambda }_a=633nm$

For an unknown liquid, the beam takes a time of $t=1.38ns=1.38\times {10}^{-9}s$

Distance travelled through the liquid,$d=30cm=0.3m$

Determination of refractive index

Speed of the beam through the unknown liquid can be determined as,

$$\begin{gathered} v=\frac{d}{t} \\ v=\frac{0.30m}{1.38\times {10}^{-9}s} \\ v=2.174\times {10}^{8}m/s \end{gathered}$$

Now, the refractive index of the liquid can be obtained by,

$$\begin{gathered} \mu =\frac{c}{v} \\ \mu =\frac{3\times {10}^{8}m/s}{2.174\times {10}^{8}m/s} \\ \mu =1.38 \end{gathered}$$

Finding the wavelength 

The wavelength of the beam through the liquid is,

$$\begin{gathered} {\lambda }_l=\frac{{\lambda }_a}{\mu } \\ {\lambda }_l=\frac{633nm}{1.38} \\ {\lambda }_l=458.7nm \end{gathered}$$